Solve the followings :
The value of the expression
`(4^n xx 20^(m - 1) xx 12^(m-n) xx 15^(m+n-2))/(16^m xx 5^(2m + n) xx 9^(m-1))` is

Options

A. 500
B. 1
C. 200
D. `1/1000`

Answer

To solve the expression

\[ \frac{4^n \cdot 20^{m - 1} \cdot 12^{m-n} \cdot 15^{m+n-2}}{16^m \cdot 5^{2m + n} \cdot 9^{m-1}}, \]

we will simplify each component step by step.

### Step 1: Rewrite the bases in terms of prime factors

1. **Rewrite \(4\), \(20\), \(12\), \(15\), \(16\), \(5\), and \(9\)**:
- \(4 = 2^2\)
- \(20 = 4 \cdot 5 = 2^2 \cdot 5\)
- \(12 = 4 \cdot 3 = 2^2 \cdot 3\)
- \(15 = 5 \cdot 3\)
- \(16 = 4^2 = (2^2)^2 = 2^4\)
- \(9 = 3^2\)

### Step 2: Substitute the rewritten bases into the expression

Substituting these values into the expression gives:

\[ \frac{(2^2)^n \cdot (2^2 \cdot 5)^{m-1} \cdot (2^2 \cdot 3)^{m-n} \cdot (5 \cdot 3)^{m+n-2}}{(2^4)^m \cdot 5^{2m+n} \cdot (3^2)^{m-1}}. \]

### Step 3: Simplify each term

Now we can simplify each term:

- **Numerator**:
- \(4^n = 2^{2n}\)
- \(20^{m-1} = (2^2 \cdot 5)^{m-1} = 2^{2(m-1)} \cdot 5^{m-1}\)
- \(12^{m-n} = (2^2 \cdot 3)^{m-n} = 2^{2(m-n)} \cdot 3^{m-n}\)
- \(15^{m+n-2} = (5 \cdot 3)^{m+n-2} = 5^{m+n-2} \cdot 3^{m+n-2}\)

Combining these, the numerator becomes:

\[ 2^{2n + 2(m-1) + 2(m-n)} \cdot 5^{m-1 + m+n-2} \cdot 3^{m-n + m+n-2}. \]

- **Denominator**:
- \(16^m = (2^4)^m = 2^{4m}\)
- \(5^{2m+n}\) remains as is.
- \(9^{m-1} = (3^2)^{m-1} = 3^{2(m-1)}\)

Combining these, the denominator becomes:

\[ 2^{4m} \cdot 5^{2m+n} \cdot 3^{2(m-1)}. \]

### Step 4: Combine the numerator and denominator

Now we can combine the numerator and denominator:

\[ \frac{2^{2n + 2(m-1) + 2(m-n)} \cdot 5^{2m+n-1} \cdot 3^{2m+n-2}}{2^{4m} \cdot 5^{2m+n} \cdot 3^{2(m-1)}}. \]

### Step 5: Simplify the powers

Now we will simplify the powers of \(2\), \(5\), and \(3\):

- **For \(2\)**:
\[ 2^{2n + 2(m-1) + 2(m-n) - 4m} = 2^{2n + 2m - 2 + 2m - 2n - 4m} = 2^{-2}. \]

- **For \(5\)**:
\[ 5^{(2m+n-1) - (2m+n)} = 5^{-1}. \]

- **For \(3\)**:
\[ 3^{(2m+n-2) - 2(m-1)} = 3^{2m+n-2 - 2m + 2} = 3^{n}. \]

### Step 6: Combine the results

Putting it all together, we have:

\[ \frac{3^n}{2^2 \cdot 5} = \frac{3^n}{4 \cdot 5} = \frac{3^n}{20}. \]

### Final Answer

The value of the expression is:

\[ \frac{3^n}{20}. \]